Van der Merwe, Lucas; Smith, Ron; Ledoan, Andrew
College of Arts and Sciences
University of Tennessee at Chattanooga
Place of Publication
The minimum rank problem is an interesting and ongoing problem in spectral graph theory which seeks to answer the question "Given a simple graph G what is the minimum rank of a matrix whose off-diagonal zero/nonzero pattern is described by G?" In recent years, the minimum rank of trees, unicyclic graphs, and cases of extreme minimum rank have been completely characterized. However, little is known about other families of graphs. Recent work in zero-forcing parameters, minimum semidefinite rank, and ranks of outerplanar graphs have given more ways to calculate upper and lower bounds for the minimum rank of a graph. We define a family of graphs with path cover number two and consider restrictions on the structure and minimum rank of these types of graphs. We consider a sub-family of these graphs and calculate the zero-forcing number and the positive semidefinite minimum rank. We also conjecture toward the minimum rank of these graphs.
M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science.
Graph theory (Mathematics); Combinatorial analysis
x, 44 leaves
Corley, Christopher M., "On the minimum rank of certain graphs with path cover number 2" (2015). Masters Theses and Doctoral Dissertations.